Given the complement of a hyperplane arrangement, let be the closure of the graph of the map inverting each of its defining linear forms. The characteristic polynomial manifests itself in the Hilbert series of in two different-seeming ways, one due to Terao and the other to Huh and Katz. We define an extension of the no broken circuit complex of a matroid and use it to give a direct Gröbner basis argument that the polynomials extracted from the Hilbert series in these two ways agree.
"A Gröbner basis for the graph of the reciprocal plane." J. Commut. Algebra 12 (1) 77 - 86, Spring 2020. https://doi.org/10.1216/jca.2020.12.77